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quaternions

Quaternion from two vector pairs

I have two vector pairs (before and after rotation).

before rotation: [x1,y1,z1] [x2,y2,z2]

after rotation: [x1',y1',z1'] [x2',y2',z2']

How to create a quaternion representing this rotation?

Solution

  • In most cases there is no rotation which transforms 2 vectors into 2 other vectors. Here is a simple way to visualize why: a rotation does not change the angle between vectors. If the angle between the 2 vectors before the rotation is different from the angle between the 2 vectors after the rotation, then there is no rotation which meets your criteria.

    This said there may be an optimal quaternion with an acceptable error which "almost" rotates your 2 vector pairs. There are a number of algorithms which vary in speed and precision to find such a quaternion. I wrote a fast C++ algorithm for an Arduino application where the speed is critical but the precision is less important.

    http://robokitchen.tumblr.com/post/67060392720/finding-a-rotation-quaternion-from-two-pairs-of-vectors

    Before rotation: u0, v0. After rotation: u2, v2.

    Quaternion q2 = Quaternion::fromTwoVectors(u0, u2);
Vector v1 = v2.rotate(q2.conjugate());
Vector v0_proj = v0.projectPlane(u0);
Vector v1_proj = v1.projectPlane(u0);
Quaternion q1 = Quaternion::fromTwoVectors(v0_proj, v1_proj);
return (q2 * q1).normalized();

    If this does not meet the requirements of your own application try to google Wabha's problem.